A Computational Study on Thermo-solute Convection in Magnetohydrodynamics (MHD) Flow With Dufour Effect
Chapter One
Aim and objectives
This research will conduct a computational study on thermo-solutal convection in MHD flow with the Dufour effect.
The objectives to attain the set aim are to:
- Examine thermal and hydrodynamic responses of the fluid to convective heat exchange parameters and also diffusion-thermo effects;
- Examine the effect of externally applied transverse magnetic field on natural convection flow in both microchannel and micro-porous channel;
- Observe the impact of heat generation parameter ( d ) on temperature, velocity profile and skin friction of the fluid;
- Investigate the impact of suction/injection on fluid velocity and skin friction on micro-porous channel flow
CHAPTER TWO
LITERATURE REVIEW
Natural Convection
Natural convection flows have in contemporary times attracted a great deal of attention due to its diverse applications in the field of engineering, industrial and heat transfer processes. In the work of Gebhart and Pera (1971), the phenomenon of natural convective flow is caused by density differences induced by temperature gradients, chemical composition gradients, and material composition. Wubshet and Makinde (2013), explained that convection cells produced from air arising above sunlight-warmed land or water are naturally a major feature of all weather systems. They explained further that convection is also seen in the rising plume of hot air from fire, oceanic currents, and sea-wind formation (where upward convection is also modified by Coriolis forces). In engineering applications, convection is commonly visualized in the formation of micro-structures during the cooling of molten metals, and fluid flows around shrouded heat dissipation fins, and solar ponds. A very common industrial application of natural convection is free air cooling without the aid of fans: this can happen on small scales (computer chips) to large scale process equipment. There are many physical problems as well as engineering applications where heat transfer by natural convection can be found to happen more frequently. They include: heat exchangers, geothermal systems, petroleum reservoirs and nuclear waste repositories, chemical catalytic reactors, packed beds, fiber and granular insulation. There have been a lot of research works on natural convection flow especially in a vertical channel under different boundary conditions studied by many authors. Gebhart and Pera (1971) studied the nature of vertical natural convection flows resulting from the combined buoyancy effects of thermal and mass diffusion. Soundalgekar and Wavre (1977) investigated heat and mass transfer effects on unsteady free convective flow along vertical porous plate with constant suction. Soundalgekar (1979) also examined the effect of mass transfer and free convection currents on flow past an impulsively started vertical plate. He observed that injecting foreign gases into the fluid flow resulted in a reduction in shear stress and mass transfer conductance. Jhaet al (2012), investigated the Natural Convection Flow of Heat Generating/Absorbing Fluid near a Vertical Plate with Ramped Temperature. They concluded that the presence of the heat sink parameter, reduces temperature and velocity profile as well as the skin friction for both ramped and isothermal cases.
Magnetohydrodynamics
The study of magnetohydrodynamics (MHD) flows have inspired great curiosity owing to its relevance in meteorology, cosmic fluid dynamics, solar physics and in the motion of Earth‟s core. On a wider perspective, applications of MHD can be appreciated in areas such as; engineering, astrophysics and geophysics particularly in geothermal energy recovery, oil extraction and thermal insulations.In the light of these, several studies on MHD natural convection flow of different fluids under different geometries have been done by many researchers. They include: Cramer and Pai (1973), Chawla (1967), Soundalgekar and Takhar (1977), and so on. Raptis and Singh (1983)studied MHD free convection flow past an accelerated verticalplate. Gupta (1961) studied the steady and transient free convection of an electrically conducting fluid from a vertical plate in the presence of magnetic field.Makinde (2009) obtained a similarity solution for the MHD boundary-layer flow and mass transfer past a vertical plate in a porous medium with constant heat flux. He found out that for positive values of the buoyancy parameters, the skin friction increased with increasing values of both the Eckert number (Ec) and the magnetic field intensity parameter (m) and decreased with increasing values of both the Schmidt number (Sc) and the permeability parameter (K). Quite recently, Jhaet al
(2014) studied MHD natural convection flow in a vertical parallel plate microchannel. They found that increase in the effects of rarefaction and fluid wall interaction led to an increase in volume flow rate while the volume flow rate decreases with increase in Hartmann number.
CHAPTER THREE
MATHEMATICAL MODELS
Flow formation and Geometry
Consider a fully developed unsteady natural convection flow of viscous, incompressible, electrically conducting fluid in a microchannel formed by two vertical plates under the influence of transverse magnetic field of uniform strength B0 which is assumed to be applied in the direction perpendicular to that of the flow. The flow is assumed to be in the direction of x¢ , which is taken vertically upward along the channel plates, and the y¢ axis is taken to be normal to the plates that are h distance apart.At the time t¢ £ 0 , the fluid is to be at rest with the initial temperature T0 and concentration C0 . At time t¢ > 0 , the temperature as well as concentration on the wall y¢ = 0 are increased or decreased to T and CP respectively where TP is the isothermal heating with temperature jump and C also has concentration jump added to the constant application of concentration on the boundary. The temperature and concentration CP on the other plate y¢ = h are maintained at T0 and C0 respectively with jumps. The geometry of the system under consideration in this present study is shown schematically in figure3.1.
CHAPTER FOUR
SOLUTIONS TO THE PROBLEMS
Solution to the Problem on Convective Boundary Condition
By introducing the Laplace transform of the dimensionless velocity, temperature and concentration equations, John (1985),
CHAPTER FIVE
RESULTS AND DISCUSSION
Convective Boundary Condition
The effect of dual-phase-lag on transient MHD natural convection heat and mass transfer flow in a vertical microchannel with convective boundary conditions was examined. Having solved the governing equations analytically in Laplace domain, the method of Riemann-sum approach is used to revert the transformed expressions to the time domain. The values obtained from the numerical inversion were presented graphically in figures 5.1.1–5.1.16 so that the influence of the different flow parameters such as Schmidt, Knudsen, Prandtl, Biotand Dufour numbers can easily be discussed. It should be noted that for the purpose of this discussion,the values of some dimensionless parameters have been defined within the ranges of 0.001 £ Kn £ 0.01, 1 £ D £ 10 , 1 £ m £ 10 and 1 £ Bi £ 10 . Meanwhile, the values of Prandtl numbers are chosen conveniently between 0.001 and 7.0 to accommodate some important fluids such as mercury, water vapour, airand water(Lienhard and Lienhard, 2006). Likewise, important diffusing species (for air) used in this analysis are ammonia (Sc=0.78), hydrogen (Sc=0.22), water (Sc=0.60) and methanol (Sc=0.97) as seen in Gebhart and Pera (1971) and Jha and Ajibade (2011).
CHAPTER SIX
SUMMARY, CONCLUSION AND RECOMMENDATION
Summary
This dissertation presented a computational study on thermo-solutal convection MHD flow with Dufour effect. The model examined the effect of dual-phase-lag on transient MHD natural convection heat and mass transfer flow in a vertical microchannel with convective boundary conditions.The problem extended further to investigate the flow characteristics in a vertical porous microchannel. Fluid motion was subjected to asymmetric heating of the channel plates with slip and jump conditions on the boundary. The solutions of the governing equations such as velocity, temperature and concentration are obtained in closed form using Laplace transform technique. Riemann-sum approximation method was then applied to invert the Laplace domain analytical solutions into time domain in order to obtain velocity, temperature, skin-friction and the rate of heat transfer. The response of mean chemical concentration, bulk fluid temperature and mass flux within the channel to different flow parameters are examined and discussed.
Conclusion
Using MATLAB software package, the hydrodynamic behaviour of the fluid flow to different flow parameters such as; Biot number (Bi), Dufour number (D), Hartmann number (m), suction/injection (S), Heat generation parameter ( d ) amongst others are presented graphically in order to have a physical understanding of the problem. It is also worth mentioning that the problem was validated numerically and an excellent agreement was established with Ajibade (2014) in the absence of Biot number, magnetic field and heat generation parameter with g = 1.
It is found that the resultant mass flux increases by growing Dufour number. Also, when the working fluid is air (Pr=0.71), massflux decreases in the presence of growing boundary convection while it increases when the working fluid is water (Pr=7.0).Mass flux is again seen to grow within the channel regardless of a small or large increase in thermal relaxation time t f. Mean concentration drops within the channel as rarefaction and concentration gradient phase lag (lC ) increase. In particular, it is observed that mean concentration of the chemical specie, ammonia (Sc=0.78) reduces due to an increase in suction/injection as well as rarefaction. Also, it is shown that mean temperature rises following an increase in Dufour (D) number but decreases with growing thermal retardation time (t T ) . In addition, diffusion-thermo (Dufour) effect wields a very significant influence on fluid temperature and velocity distribution. Furthermore, the presence of heat generation parameter (H), increases temperature, velocity profile and also skin friction of the fluid. Finally, the magnetic field effect is to decrease the fluid velocity within the microchannel.
Recommendations
This present research work which centres on the computational study of thermo-solutal convection MHD flow with Dufour effect has its many applications in the advancement of the frontiers of science and technological innovations particularly in meteorology, astrophysics, geophysics, design of nuclear reactors, petroleum reservoirs etc. This present study however could be extended to capture what impact magnetic field, heat source as well as convective heat exchange parameter could have on the fluid flow when the microchannel is oriented horizontally. This dissertation considered MHD flow of electrically conducting fluid with applied magnetic field. It can however be reconsidered in the presence of induced magnetic field.
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